Lie Ideals in Associative Algebras

1984 ◽  
Vol 27 (1) ◽  
pp. 10-15 ◽  
Author(s):  
G. J. Murphy
Keyword(s):  
Lie Ideal ◽  
Associative Algebras ◽  
Extensive Class ◽  
Simple Algebras ◽  
Associative Rings

AbstractIt is shown that in a certain extensive class of algebras one can associate with each Lie ideal a corresponding associative ideal which facilitates the study of Lie ideals, especially for simple algebras. We apply this construction to obtain new, simpler proofs of some known results of Herstein [10] and others on the Lie structure of associative rings.

Varieties satisfying semigroup identities: Algebras over a finite field and rings

2016 ◽  
Vol 26 (05) ◽  
pp. 985-1017
Author(s):  
Olga B. Finogenova
Keyword(s):  
Finite Field ◽  
Associative Algebras ◽  
Adjoint Semigroup ◽  
Semigroup Identities ◽  
Associative Rings

We study varieties of associative algebras over a finite field and varieties of associative rings satisfying semigroup or adjoint semigroup identities. We characterize these varieties in terms of “forbidden algebras” and discuss some corollaries of the characterizations.

scholarly journals Some remarks on Mathieu subspaces over associative algebras

2017 ◽  
Vol 16 (05) ◽  
pp. 1750089
Author(s):  
Michiel de Bondt
Keyword(s):  
Commutative Rings ◽  
Associative Algebras ◽  
New Concepts ◽  
Simple Algebras ◽  
Algebraic Elements

In [9] (which is [W. Zhao, Mathieu subspaces of associative algebras, J. Algebra 350(1) (2012) 245–272]), the author takes a closer look at algebraic elements of radicals of Mathieu subspaces (of associative algebras) over a field, and suggests to look at integral elements with rings other than fields. But it seems more useful to look at so-called co-integral elements. We generalize his theory about algebraic radicals over fields to co-integral radicals over commutative rings with unity. Furthermore, we show that over Artin rings, the concepts of integrality and co-integrality coincide. In addition, we define so-called uniform Mathieu subspaces, inspired by the fact that Mathieu subspaces with co-integral radicals are always of this type. Besides broadening the theory of [9] by means of the new concepts co-integrality and uniformity, we generalize many of the results of [9] in other ways as well. Furthermore, we obtain several new results. In the last section, we disprove a conjecture by the author of [9] (in a version of [9] prior to finding the counterexample), by showing that so-called strongly simple algebras do not need to be fields over theirselves.

Associative rings in which nilpotents form an ideal

2019 ◽  
Vol 56 (2) ◽  
pp. 177-184
Author(s):  
Orest D. Artemovych
Keyword(s):  
Unit Group ◽  
Lie Ideal ◽  
Nilpotent Elements ◽  
Associative Rings ◽  
Torsion Free

Abstract It is shown that if N(R) is a Lie ideal of R (respectively Jordan ideal and R is 2-torsion-free), then N(R) is an ideal. Also, it is presented a characterization of Noetherian NR rings with central idempotents (respectively with the commutative set of nilpotent elements, the Abelian unit group, the commutative commutator set).

scholarly journals On Lie Structure in Semiprime Inverse Semirings

2019 ◽  
pp. 2711-2718
Author(s):  
Rawnaq KH. Ibraheem ◽  
Abdulrahman H. Majeed
Keyword(s):  
Lie Ideal ◽  
Definition Of ◽  
Associative Rings

In this paper we introduce the definition of  Lie ideal on inverse semiring and we generalize some results of Herstein about Lie structure of an associative rings to inverse semirings.

New Simple Algebras Containing the Matrix Ring

2011 ◽  
Vol 18 (spec01) ◽  
pp. 775-784
Author(s):  
Jongwoo Lee ◽  
Ki-Bong Nam
Keyword(s):  
Automorphism Group ◽  
Matrix Ring ◽  
Matrix Group ◽  
Finite Dimensional Algebra ◽  
Associative Algebras ◽  
Dimensional Algebra ◽  
General Matrix ◽  
Finite Dimensional ◽  
The Matrix ◽  
Simple Algebras

We define the combinatorial simple algebras N(eAS,n,t)k, N(eAS,n,t)k+and N(eAS,n,t)[k]which contain both the matrix ring and the non-associative algebras discussed in [11]. We also define a new cyclic finite dimensional algebra which contains the matrix ring Mn(𝔽) and show that the algebra is simple (see [3] and [7]). Even more, we find the automorphism group of the algebra N(0,n,t)[1]and show that the general matrix group GLn+t(𝔽) is not a subgroup of the automorphism group Aut (N(0,n,t)[1]).

Coordinatization Theorems For Graded Algebras

2002 ◽  
Vol 45 (4) ◽  
pp. 451-465 ◽  
Author(s):  
Bruce Allison ◽  
Oleg Smirnov
Keyword(s):  
Bilinear Form ◽  
Simple Algebra ◽  
Associative Algebras ◽  
Graded Algebras ◽  
Morita Theory ◽  
Coordinate Algebra ◽  
Simple Algebras ◽  
One To One ◽  
Simple Ideal

AbstractIn this paper we study simple associative algebras with finite -gradings. This is done using a simple algebra Fg that has been constructed in Morita theory from a bilinear form g : U × V → A over a simple algebra A. We show that finite -gradings on Fg are in one to one correspondence with certain decompositions of the pair (U, V). We also show that any simple algebra R with finite -grading is graded isomorphic to Fg for some bilinear from g : U × V → A, where the grading on Fg is determined by a decomposition of (U, V) and the coordinate algebra A is chosen as a simple ideal of the zero component R0 of R. In order to prove these results we first prove similar results for simple algebras with Peirce gradings.

Group Gradings on Associative Algebras with Involution

2008 ◽  
Vol 51 (2) ◽  
pp. 182-194 ◽  
Author(s):  
Y. A. Bahturin ◽  
A. Giambruno
Keyword(s):  
Abelian Group ◽  
Matrix Algebra ◽  
Finite Abelian Group ◽  
Closed Field ◽  
Base Field ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
The Matrix ◽  
Simple Algebras ◽  
Group Gradings

AbstractIn this paper we describe the group gradings by a finite abelian group G of the matrix algebra Mn(F) over an algebraically closed field F of characteristic different from 2, which respect an involution (involution gradings). We also describe, under somewhat heavier restrictions on the base field, all G-gradings on all finite-dimensional involution simple algebras.

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